Find a simplified expression for cos(tan^(-1)(x/5))
Evaluate cos(sin^(-1)(1/11)), giving your answer as an exact value (no decimals)
draw the triangles.
second one: sides of triangle are 1,sqr120, 11(Hypotenuse)
cosine theta=sqrt120/11
first one: sides of triangle are -1 (opposite side), x/5, and hypotensuse is sqrt (1+x^2/25)
cosTheta=x/5 / sqrt(1+x^2/25)
bobpursley can u help with mine
Thank you so much!!!!!
To find a simplified expression for cos(tan^(-1)(x/5)), we can use the identity that relates tangent and cosine:
tan^(-1)(x) = arctan(x)
Using this identity, we can rewrite the expression as:
cos(tan^(-1)(x/5)) = cos(arctan(x/5))
To simplify this expression, we need to use the identity that relates cosine and arctan:
cos(arctan(x)) = 1 / √(1 + x^2)
Applying this identity to our expression, we have:
cos(arctan(x/5)) = 1 / √(1 + (x/5)^2)
Simplifying further, we get:
cos(arctan(x/5)) = 1 / √(1 + x^2/25)
Therefore, the simplified expression for cos(tan^(-1)(x/5)) is 1 / √(1 + x^2/25).
Now let's evaluate cos(sin^(-1)(1/11)) to get an exact value.
To evaluate this expression, we can use the Pythagorean identity that relates sine, cosine, and the square root. The Pythagorean identity is:
sin^2(theta) + cos^2(theta) = 1
Using this identity, we can rewrite the expression as:
cos^2(sin^(-1)(1/11)) = 1 - sin^2(sin^(-1)(1/11))
Since sin^(-1)(1/11) represents an angle whose sine is 1/11, we can substitute sin^(-1)(1/11) with x. This gives us:
cos^2(x) = 1 - sin^2(x)
Now, we know that sin(x) = 1/11. So, we can use this information to solve for cos^2(x):
cos^2(x) = 1 - (1/11)^2
cos^2(x) = 1 - 1/121
cos^2(x) = 120/121
To find the value of cos(x), we take the square root of both sides:
cos(x) = ± √(120/121)
Since cosine is positive in the first and fourth quadrants, we take the positive square root:
cos(x) = √(120/121)
Therefore, the exact value of cos(sin^(-1)(1/11)) is √(120/121).